**Find the values of the letters in each of the following and give reasons for the steps involved.**

1.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We can observe that A+5 gives 2 as the unit's digit to the sum.So it's only possible if A=7.

Now, the sum becomes :

And we can clearly see that value of B=6.

Therefore we got the values as A=7 and B=6

2.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We can observe that A+8 gives 3 as the unit's digit to the sum.So it's only possible if A=5.

Now, the sum becomes :

By this, we can clearly see that value of B and C as 4 and 1 respectively.

Therefore we got the values as A=5, B=4 and C=1

3.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We can observe that A×A gives A as the unit's digit to the multiplication, that means it's only possible if A=1, 6 and 5.

Checking for all the case:

For A=1:

We can see clearly that it is not similar with the given multiplication so A

For A=5:

We can see clearly that it is not similar with the given multiplication so A

For A=6:

We can see clearly that it is similar with the given multiplication so A=6.

4.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We have here two conditions :

(i) B+7=A

(ii)A+3=6

By solving the second equation we get A=6-3=3

Also if we keep A=3 directly in eq. (i) we get B=3-7=negative value. So it can be said that a carry was taken so let the carry be 1.Therefore, now A=2+1(carry) and let us check if eq. (i) satisfies it or not.

We have B+7=12[? 1(carry)2(value of A)]

So we get B=12-7=5.

So the sum becomes :

Hence, the assumption is correct and values are A=2 and B=5

5.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We can observe that we have equations:

(i)B

Solving it we can have B=0 or 5

5 is not taken because, for satisfying the condition A

(ii)A

we can satisfy this condition only if A=5 i.e., 5

So we have A=5, B=0 and C=1

6.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We can observe that we have equations:

(i)B

Solving it we can have B= 0 or 5

5 is removed because for satisfying the condition A

(ii)A

we can satisfy this condition only if A=5 i.e., 5

So we have A=5, B=0 and C=2

7.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We can observe that: Possible values of BBB are 111, 222, 333, etc.

Let us divide these numbers by 6 and check the nummber which is completely divisible.

111 ÷ 6 =18, remainder 3 so, 111 is rejected.

222 ÷ 6 = 37, remainder 0 but the quotient 37 is not of the form A 2. So,222 is rejected.

333 ÷ 6 = 55, remainder 3 so, 333 is rejected.
444 ÷ 6 = 74, remainder 0 and also the quotient 74 is of the form A 4.

So, we can say the number B=4 is satisfying.Let's check and find value of A

We know this is only possible if A=7

Hence, we have got A=7 and B=4

8.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We have condition as:

B+1=0 i.e, B must be 9 to get 0 at unit's digit of the addition.So, we got B=9

Also we have A+1+1(carry)=B

Hence, we have A=7 and B=9

9.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We have condition as:

B+1=8.So, we got B=7

Also we have A+B=1

A must be 4 to get unit's digit of A+7 as 1.So, A=4

And the last condition we have 2+A+1(carry)=B

Checking the values of A and B if it satisfies or not:

We have A=4 and B=7.Substituting, we have:

LHS: 2+A+1=2+4+1=7

RHS: B=7

Hence the values are verified.So we have A=4 and B=7

10.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We have condition as:

(i)A+B=9.

(ii)2+A=10.

(iii)1+6+1(carry)=A

We can say by equation (iii) and equation (ii) that A=8

Also, we have A+B=9. Sustituting A=8, we get B=1

Hence, we have A=8 and B=1

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

For a number is divisible by 9 we know that the sum of its digits should also divisible by 9.

Here, Sum of the digits of 21y5 = 2 + 1 +y + 5 = 8 + y

So we should have (8 + y) ÷ 9 = 1

Hence, the required value of y = 1.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We know that a number is a multiple of 9 if the sum of its digits is also divisible by 9.

Here, Sum of the digits of 31z5 = 3 + 1 + z + 5

So we have , 3 + 1 + z + 5 = 9k where k is an integer.

Checking for different values of k

For k = 1:

For k = 2:

We have, 3 + 1 + z + 5 = 18

k = 3 is not possible because 3 + 1 + z + 5 = 27

Hence the required value of z is 0 or 9

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We are given that 24 x is a multiple of 3. So, the sum of its digits 2 + 4 + x = (6 + x) should be a multiple of 3.

Therefore, (6 + x) is one of the numbers 0, 3, 6, 9, 12, 15, 18…so on but as per given x can only be a digit. Therefore, (6 + x) must be equal to 6 or 9 or 12 or 15.

i.e., 6 + x = 6 or 9 or 12 or 15

Thus, x can have any of the four different values, namely, 0,3,6 or 9.

NCERT Solutions for Class 8 Maths Chapter 16 Playing With Numbers

Answer :

We are given that 31z5 is a multiple of 3. So, the sum of its digits 3 + 1 + z + 5 = (9 + z) should be multiple of 3.

Therefore, (9 + z) is one of the numbers 0, 3, 6, 9, 12, 15, 18, …so on but given that z is a digit. Therefore (9 + z) must be equal to 9 or 12 or 15 or 18.

So we have, 9 + z = 9 or 12 or 15 or 18

Hence, z can have any of the four different values, which are 0, 3, 6 or 9.

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